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Turbulent Combustion:Modelling and Applications
Princeton Summer School 2021
E. Mastorakos
Engineering Department
University of Cambridge
1
Structure of lectures
Day 1: The context
Introduction to combustion devices; gov. eqtns and the fundamental problem of
turbulent combustion; classifications; “bird’s eye” view of modern turb. comb.
research
Day 1-2: Revision of basics
Some fundamental aspects of turbulence and mixing; a revision of some laminar
flame problems
Day 3: Modelling & applications
Models: infinitely fast chemistry (EBU, ED etc); “laminar chemistry”; non-premixed
& premixed flames (flamelet, CMC, PDF); other models for premixed
Day 4: Modelling & applications
Diesel and natural gas engines; autoignition; gas turbine combustion; extinction;
soot and NOx
PLEASE INTERRUPT AND ASK QUESTIONS!
Main education targets:
(i) To build physical intuition;
(ii) To develop skills in turb combustion modelling;
(iii) To be able to put your own research in context.
2
Bibliography
Books:
“An Introduction to Turbulent Reacting Flows”, Cant & Mastorakos, ICP, 2007Introductory level; background; basics on the various models
“Turbulent Combustion Modelling”, Echekki & Mastorakos (Eds), Springer, 2011Advanced level; revision of major modern models & techniques
“Theoretical and Numerical Combustion”, T. Poinsot & D. Veynante, R.T. Edwards, 2005Fundamental combustion, good introduction to turbulent combustion
“Turbulent Combustion”, N. Peters, Cambridge University Press, 2000Flamelet model, turbulent premixed flame speed
Progress in Energy and Combustion Science (journal):
“Turbulent combustion modelling”, D. Veynante, L. Vervisch, PECS 28 (2002) 193-302.Theory and connections between models
“Turbulent premixed combustion: Flamelet structure and its effect on turbulent burning
velocities”, J. Driscoll, PECS 34 (2008) 91-134.Focus on premixed flame structure and models
“Ignition of turbulent non-premixed flames”, E. Mastorakos, PECS 35 (2009), 57-97.Focus on ignition (application to diesel and g.t. engines)
“LES of gaseous flames in gas turbine combustion chambers”, Gicquel et al., PECS 38
(2012) 782-817.Focus on thickened flame model, CFD approach, thermoacoustics
“Modeling of turbulent dilute spray combustion”, Jenny et al., PECS 38 (2012) 846-887.
Focus on sprays, models for range of combustion modes
3
Day 1: Introduction and Statement of the Problem
Main applications: WHY do we bother?
Governing equations: WHAT is the problem?
Flame classification
Major issues in modern turbulent combustion research
International Workshops
4
Primary energy sources - worldwide
Source: IEA Statistics
COAL
OIL
GAS
NUCLEAR
BIOMASS
SOLAR,
WIND,
GEOTH.
HYDRO
CO
MB
US
TIO
N
5
Primary energy sources - worldwide
Source: IEA Key World Energy Trends 2016
Around 90% comes from
combustion (coal, oil,
natural gas, biofuels)
6
Fossil fuel’s use for electricity
COAL
OIL
GAS
NUCLEAR
SOLAR, WIND,
GEOTHERMAL,
WASTE HYDRO
CO
MB
US
TIO
N
Source: IEA Statistics
7
Source: IEA Statistics
Coal’s contribution is reducing
8
Machines that use fuels
Liquid
(diesel,
kerosene,
SAF)
Coal
and
solid
fuels
Spray autoignition; o(10-100kW) Wartsila marine engine (~80MW)
In-furnace view of flame
o(100MW)
Nat Gas,
(H2, NH3)
Drax power station, UK
o(1GW), biomass
Hopkinson Lab (1bar)
o(10kW)GE/Alstom GT26; o(300MW)
o(10m)
o(0.1m)
Aeroengine; o(1MW)
per combustor
o(200MW) take-off
9
“With some 100,000 years of development, [combustion] might be expected
to be a mature technology. In fact, it is the least developed technology of
modern engineering systems. Engineering systems with mature technology
have computer models that are routinely used in analysis and can be used in the
optimization of the geometry and operating conditions of new designs. Thus the
wings of new models of jet aircraft have subtle geometry contours that greatly
increase their efficiency. The same can be said for the new generation of fans in
turbo-fan engines. Computer modeling of combustors is still at a fledgling stage
and is only used peripherally in the development of new combustors. Combustor
development is still largely by cut-and-try testing in experimental rigs and in
prototype and in-service engines and power plants”.
R. W. Bilger, “The role of combustion technology in the 21st century”, in “Turbulent
Combustion Modelling”, Echekki & Mastorakos (Eds.), Springer, 2011.
10
Combustion: an area where application is ahead of the scientific understanding
Many different phenomena (chemistry, heat transfer, fluid mechanics, multi-scale, surface
reactions, pollutants, large range of chemical timescales etc)
Huge range of length scales (0.01m – 10m)
Huge range of residence times (ms to s)
Huge range of power (1kW to 100MW)
Some of this range is taken care of by the high pressure (more mass flow per unit time and
area); but the remaining (many orders of magnitude) is achieved by the turbulence.
Fundamental effect of turbulence on reaction:
As Reynolds number increases, range of eddy length- and time-scale increases, hence
more flame area, hence more reaction rate per unit volume.
Question: why does a spark ignition engine work at both 600 and 6000 rpm?
But:
If residence time too short, combustion not complete. Finite-rate kinetics!
Also:
Even if combustion OK, how about pollutants (CO, NOx, soot)? Finite-rate kinetics!
Legislation drives technological innovation; but theory lags behind.
11
Combustion in a low-carbon future?
Aviation: difficult to see medium- and long-range airplanes with electric motors (need high
energy storage with high kJ/kg); current focus on H2 (short/medium range) & SAF (long-
range).
Shipping: ~ similar to aviation; focus on NH3? CH2OH? H2? Biofuels?
Road transport (passenger cars): easier to fully electrify.
Road transport (heavy goods): some use of fuels (H2? Biofuels?) likely.
Electricity production: coal is on its way out; natural gas will increase.
CCS: pre- or post-combustion?
Accidents, explosions, nuclear safety, fire safety: combustion science necessary
Battery fires!
E-fuels & biofuels.
Studying combustion (kinetics, heat transfer, fluid mechanics, multi-scale phenomena,
laser-based methods, CFD, etc) helps develop insights for many other sectors.
12
Future CO2 stabilisation scenarios
(“450”=450 ppm CO2)
Source: IEA statistics ; TPES: Total Primary Energy Supply
COAL
OIL
GAS
13
Future of combustion?
Even when we replace fossil fuels with biomass-derived or e-fuels (renewable electricity to
fuels), high-T oxidation processes will be relevant. We need to develop technologies that:
• Offer higher fuel economy (i.e. reduce CO2 per unit of output)
• Reduce other pollutants (NOx, SOx, particulates)
• Facilitate carbon capture and storage (pre- and post-combustion)
• Offer fuel flexibility (low-LCV fuels, various NGs, various oils etc)
• Promote e-fuels (Hydrogen, ammonia, methanol, other HCs)
• Are safe (fires, explosions)
• Most of these applications have high Reynolds number
• Turbulent combustion remains important!
• We must engage more with the fuel producer.
14
Governing equations – the fundamental problem 1
Combustion is governed by a set of equations expressing the principles of conservation of
overall mass, momentum and energy (“Fluid Mechanics” & “Heat transfer”) and mass of
individual species (“Mass transfer”), supplemented with a set of closures for heat and mass
fluxes and for chemical source terms (“Chemical kinetics”). They are reproduced below
(from Echekki & Mastorakos, 2011) for demonstration:
Mass
Momentum
Species k
Total internal energy
Closures
15
Governing equations – the fundamental problem 2
For demonstration, assume 1-step chemical reaction (Sp1 + Sp2 = Sp3). The chemical
source term w is written as:
Observation 1: If Tact high, the reaction rate increases very steeply with
temperature. The steep dependence of the reaction rate on the temperature is
one of the most important aspects of combustion!
Observation 2: Species 1 and 2 are coupled; consuming one means consuming
the other. Chemistry develops connections between the source terms of the
various species. Never forget this!
16
Governing equations – the fundamental problem 3
In turbulent flows, we need averaging. Consider time- or ensemble-averaging first
(same comment applies to LES filtering)
(assume constant density for ease of presentation)
Turbulent flux of species due to velocity fluctuation in i-direction (needs
modelling)
Observation 3: Evaluating mean reaction rate at the mean Y and mean T is very wrong!
17
( )( )
+−++=
TT
TYYYYAw act
BBAAC exp
−
T
TYYAw act
BAC exp
),,...,,( general,In 21 TYYYfw N
The fundamental problem 4 – example of thin premixed flame in CFD
CFD cell, cube of side L
Reaction zone,
thickness dR
Preheat
zone
Burnt
zone
T=2100K
Fuel
T=300K
If dR<<L, and ½ of cell is reactants, the ½ of cell is
products:
volume averaged T, <T> = 1200K
(and volume averaged Yfuel=YF,0/2)
If, say, Tact=20000K (typical hydrocarbon fuel
value), exp(-Tact/<T>)=5.8E-8
But <exp(-Tact/T)>=[exp(-Tact/300)+exp(-
Tact/2100)]/2=7.3E-5
Three orders of magnitude error!
YF,0 General observation 4: The average of a non-
linear function IS NOT equal to the function
evaluated at the average value of its arguments.
<f(x)> ≠ f(<x>) if f(x) is nonlinear
18
The fundamental problem – conclusion
Therefore, in turbulent combustion CFD, we never evaluate the reaction rate at the values of
species mass fractions and temperature we are solving for.
Unless we are doing DNS, we need a MODEL for the mean reaction rate!
In LES, the sub-grid fluctuations are smaller, but model is still needed.
If LES reaches DNS resolution (as in some recent lab-scale simulations), the absence of a
sub-grid combustion model is not catastrophic. But to generalise good agreement from DNS-
like simulations of lab-scale flames to practical situations is wrong!
The Arrhenius term is driving turbulent combustion research. We do not yet have a fully
predictive theory.
Practical devices are virtually always turbulent – we rely on the turbulence to get high
reaction rate per unit combustor volume.
19
Flame classification and range of quantities of interest
Lab-scale (~10kW), swirling n-heptane spray flame (Marchione et al., CNF 2009):
Blue or yellow; long or short; stable or unstable: depending on mass flow rate of
air, fuel, spray pattern, swirl number, combustor diameter etc…
Large differences in behaviour!
20
Flame classification
Premixed flame,
stabilised on round
disk. Model for
afterburner or natural-
gas gt flame.
Series of swirling premixed
flames arranged in a circle,
mimicking annular combustor.Heptane spray flame in
swirling air. Streamline
sketch typical of
burners.
Source: Mastorakos’s own research
21
Primary & secondary
atomization, dispersion
Chemical mechanism,
turbulence-chemistry
interactions
Heat loss,
radiationNOx, CO, soot,
Temperature
profile, velocity,
vorticity
Flame location Aerodynamics
Thermo-acoustics
(coupling with combustor
acoustics)RED: Usual targets
BLUE: Models
Heat transfer
Example: phenomena present in gas turbine combustors
22
Designing a combustor: OK, not quite a Herculian task, but still…
23
Interesting mythological trivia:
Hercules finally killed Hydra by bringing along his young nephew to apply *fire* in every head he was chopping off so new ones would not pop-up…
Soot
NOx
Blow-off
Thermoacoustics
PhD student or engine
manufacturer designing a new
combustor…
Ignition
Heat
transfer
Flashback
Little by little combustion science helps
24
1960s 2010s B787
Q: Is this improvement solely due to more knowledge on soot, or also due to better
understanding of flame stabilization and spray?
Flame classification – wide spectrum!
“Premixed”: fuel + air fully mixed before flame
“Non-premixed”: fuel and air arrive at flame from separate sides; little overlap
“Stratified”, “Partially-premixed”: a mix between the two; propagation across a range of equivalence ratio
Spray flames: elements of all three above
Flames in gas turbines: all types can be present at different places, conditions, designs
Flames in diesel engines: non-premixed (and autoignition); more premixing in recent Low-T Combustion concepts
Flames in spark-ignition engines: premixed or stratified (and spark ignition)
Pollutants: some pollutants created post-flame; hence concept of well-stirred or imperfectly-stirred reactors and plug-flow reactors useful
“How much chemistry do we need in a real combustor CFD?”
25
Flame classification – wide spectrum!
“Edge flames”:
- Probably best understood by
non-premixed flame concepts
- Some recent work with DNS &
experiments
Mixture fraction
Dis
tan
ce n
orm
al to
pro
pag
atio
n
“Stratified flames”:
- Probably best understood by
premixed flame concepts
- Work picking-up everywhere
“Fully Premixed”
“Non-Premixed” (no
edge, no propagation)
Rele
van
t to
sp
ray
xrichxlean xst
Mixture fraction x: to be discussed later
For now: equal to fuel distribution, if no
combustion
26
Laminar vs. turbulent combustion
Laminar burning velocity is the outcome of reaction/diffusion balance.Fundamental property of a fuel/oxidiser mixture.
Heat release rate per m2: rYfuSL(LHV).
Source: Dibble, Mass & Warnatz, “Combustion”, Springer-Verlag, 1996
SL=Vsinq
q
V
SL
27
Laminar vs. turbulent combustion
Turbulence wrinkles and stretches flame and therefore increases flame surfacearea, therefore quicker burning. “Turbulent flame speed” ST depends on u’ and SL,mostly. But too much u’ can kill flame. Very intensive research at present.
Source: Shy et al., Proc. Roy Soc A (2000) 456:1997-2019
ST = <V> sinq
q
<V>
ST
28
Modern turbulent combustion research
Wide spectrum of people, approaches, backgrounds, focal points.
More powerful computers ➔ LES in industry becomes common.
No model is fully OK for all combustion phenomena!
Close collaborations between experiment and modelling.
Key questions preoccupying turbulent combustion researcher and practitioner:
“How much chemistry do I need in a real combustor CFD?”
“Which canonical combustion problem best represents my turbulent flame?”
“What makes my research publishable in CNF these days?”
29
Trends in modern turbulent combustion research
International Workshops:
- Turbulent Non-premixed Flames
- Used to be targeting finite-rate kinetics (local extinction) in non-prem; now more on stratified
flames. Piloted or bluff-body stabilised with two streams of unequal f. Lots of attention to model
validation
- Turbulent Premixed Flames
- Focused scientific discussions; more on physics, less on model success.
- Turbulent Combustion of Sprays
- Piloted jet flame (Sydney), spray in co-flow (Delft, Rouen), swirl spray flame (Cambridge)
- Focus on model validation (including evaporation)
- Sooting Flames
- Focus on laminar flames
- Some turbulent target flames (DLR, Adelaide)
- Do we need one for NOx? For NH3? For H2?
- International Symposium, regional conferences: more on fundamentals
- AIAA, ASME, SAE: more on applications
- Finite-rate kinetics, pollutants, flame structure still important points of research
- VERY IMPORTANT: dialogue between modellers and experimentalists is needed!
30
Trends in modern turbulent combustion research
DNSExperiment
RANS/LES
Low-order physical
modelling
Our best estimate
of the truth!
Useful codes
and intuition
for realistic
devices
31
Not one-way information route; everybody learns from everybody in turbulent combustion
Trends in modern turbulent combustion research
Careful: “If you are holding a hammer, you think everything is a nail”…
OK, but not everything is turbulent:
True. We need to follow laminar flame and chemistry developments.
LES vs RANS:
Both are used in industry; LES in research. Industry catches up.
Accuracy vs. cost; how many LES can industry do for design?
Lots vs. little chemistry:
For flame position, perhaps simple chemistry is OK (except autoignition); for pollutants and
extinction, chemistry of some complexity needed. Current experience: 50-100 species in
chemical mechanism.
Solve chemistry on-line (e.g. CMC, PDF, LEM, TFM) or use pre-tabulation (e.g. flamelet)?
Comprehensive vs. approximations:
Do we need to include complex evaporation, radiation, high-Ma etc in ALL simulations?
32
Conclusions – Day 1 (Context)
Practical applications of combustion: high Re
High Re instrumental for enabling large range of power from combustion devices
Combustion will be with us for decades
No model works for everything
Validation focused on physical phenomenon still needed (e.g. for blow-off, ignition, pollution
etc)
LES becomes standard
Collaborations needed
A researcher in turbulent combustion is always under attack – “not enough chemistry” for
the chemist; “not enough fluids” for the fluid mechanician; “not practical enough” for the
industrialist…But: somewhere within there you get a useful answer – if you understand the
key limitations, trends and dependencies.33
Turbulent Combustion:Modelling and Applications
Princeton Summer School 2021
E. Mastorakos
Engineering Department
University of Cambridge
1
Day 2: Background on turbulent mixing and some laminar flame phenomena
Turbulence modelling
Mixing and scalar dissipation
Autoignition
Well-stirred reactor
Laminar premixed and non-premixed strained flames
Heat loss vs. chemistry
2
Statistical description
Randomness: we cannot say with certainty that an event will or will not occur. For example, that “u(t) will be between 10 and 11 m/s”.
Probability density function
Possible timeseries of a turbulentsignal (e.g. u, T)
u’
u : instantaneousu’ : fluctuationu : average
Time average(for stationary flow)
Ensemble average(for non-stationary)
3
P(r) : probability that r < u < r+dr
Properties: P(r) > 0 ; 𝑎𝑏𝑃 𝑟 𝑑𝑟 = 1 ;
r takes values only in [a,b] ; [a,b] is called the sample space
Probability that 𝑣1 < 𝑢 < 𝑣2 = 𝑣1𝑣2 𝑃 𝑟 𝑑𝑟
In turbulence, P(u) nearly Gaussian.
For velocity, [a,b]=[-,]
For scalar, [a,b] bounded; P(Temperature) could have funny shapes…
4
Multivariate PDF:
Gives us the statistics of two quantities (for example: the velocities in two directions), and their relationship (in a statistical sense).
P(u,v) = probability that u1<u<u2 AND v1<v<v2
Properties: P(u,v) > 0 ; 𝑎1𝑏1𝑎2𝑏2𝑃 𝑢, 𝑣 𝑑𝑢 𝑑𝑣 = 1
5
Averaging with the PDF:
Mean of u and of a function f(u) and the variance s2 are given by:
6
In turbulence parlance, the variance is the average of the squared fluctuations
Typical iso-probability contours:
7
Correlation coefficient:
Energy cascade & eddy scales
8
A. Turbulence has a large range of eddy sizes.B. Large ones “break-up” to smaller ones due to instability.
A + B = “energy cascade”
Kinetic energy flows from large eddies to small ones, eventually the process stops due to viscosity.
Largest
Smallest
Lturb
hK
Energy cascade & eddy scales
9
Hot
Cold
Large scale Small scale
(Note: these images are not truly consistent, they are not from the same experiment, they are usedfor illustrating the concept.)
Velocity, length, and timescales : large-eddy scales
Turbulent kinetic energy k : 𝑘 =1
2𝑢′𝑖𝑢′𝑖 =
1
2𝑢′1𝑢′1 + 𝑢′2𝑢′2 + 𝑢′3𝑢′3
Typical or “characteristic” turbulent velocity fluctuation u : 𝑘 =3
2𝒖2
Lturb : size of large eddies = o(width of flow)
Tturb = Lturb / u : eddy turnover time
eddy lifetime
integral timescale
10
Vortex stretching:
The fundamental mechanism by which eddies break apart, vorticity gets concentrated in patches, turbulence is 3D, small-scales become isotropic etc
Conservation of angular momentum in inviscid, incompressible fluid:
11
w1
w2>w1
Instability
w3>w2
Vortex-induced flow causes further stretching
“Spaghetti”
What is an “eddy”?
12
Vorticity is concentrated in tubes or sheets; with quiet fluid around them.“3-D Spaghetti” model.
High scalar & kinetic energy dissipation regions highly localised.
Turbulence modelling
Governing equations for mean U or scalar have higher-order correlations.
Turbulent scalar fluxes and Reynolds stresses need closure. Usual closure: gradient
hypothesis
Important concept: energy cascade and kinetic energy dissipation
13
Kolmogorov scales
(for simple shear)
Dissipation (u2)/(eddy lifetime = Lturb/u)
Turbulence modelling
k-e turbulence model: the workhorse in industry, initial condition for LES etc.
Do not alter “constants”; if it does not work, use another model (Reynolds stress, LES etc)
14
Turbulent mixing – molecular chaos / random walk
15
Continuum (Eulerian) vs. “particle” (Lagrangian) approach: Do k jumps per unit time,
each a distance Dx, in random direction; probability that particle will be in position
m is:
This is the solution to the 1-D diffusion equation with diffusivity D=(kDx)Dx/2 ; (kDx) can be
understood as a characteristic velocity.
Basic idea behind Brownian motion, random walks, PDF method etc.
Turbulent mixing – large scale dispersion
16
Lagrangian approach: “Turbulent diffusion by continuous movements” (G.I. Taylor
1922).
x
y
zV
X
In 1-D for simplicity:
Lagrangian autocorrelation coefficient
Lagrangian integral timescale
RL(t)
t
Turbulent mixing – large scale dispersion
17
Short times:
Long times:
So, for
So, for
Turbulence has “memory”. It is not like white noise.
Distinction between short-time and long-time mixing needed often in turbulent combustion:
(i) growing flame kernel experiences only a range of eddies;
(ii) growth of V-flame brush;
(iii) attachment points vs. far-field regions;
(iv) attention to fuel inlets needed.
Turbulent mixing – scalar dissipation
18
Above equations are for mean and variance of a passive scalar (i.e. no source term).
Expressed in density-weighted averages.
Gradient approximation has been used: not good in general for complex flows, hence the
motivation for LES.
Scalar dissipation: very important quantity in turbulent combustion. Appears in premixed &
non-premixed models.
Usual model:
scalar dissipation = (energy to be dissipated)/(eddy timescale)
”Energy cascade” idea for the scalar.
Constant Cd to make variance decay agree with experiment. Usually, Cd=2 is OK. For
early mixing problems,Cd>2 (Merkides & Mastorakos, Chem Eng Sci, 2005).
Modelled
mean scalar dissipation
−𝐷𝑡𝑢𝑟𝑏𝜕 ሚ𝜉
𝜕𝑥𝑘
𝑔2 = ෪𝜉′2
2෩𝑁 ≡
෫
2𝐷𝜕𝜉
𝜕𝑥𝑖
2
Turbulent mixing – scalar dissipation
19
Small-scale intermittency of velocity field leads to filament-like structure of scalar
dissipation field. Smallest scale is the Bachelor scale:
22
+
dy
d
dx
d
PLIF of tracer in air
Turbulent mixing – scalar dissipation
20
Conditional scalar dissipation appears often in models (will be discussed later)
Turbulent mixing – probability density function of scalars
21
Instantaneous
Mean
Scalar PDF is bounded [0,1]
Turbulent mixing – probability density function of scalars
22
Inertial range
23
Question: Can I have some universal theory for the energy distribution across eddy size?
Answer: Kolmogorov said “yes”, under certain conditions.
Assumptions: (i) dissipation independent of scale (again); (ii) viscosity is important only at small scales (again). Hence: 𝐸 𝜅 = 𝑓(𝜅, 휀).
Units: k: m-1 ; E: m3s-2 ; e: m2s-3. 3 quantities, 2 units, therefore:
Very strong experiment evidence for the “-5/3 law” for all flows. One of the “footprints” of a turbulent signal.
Energy spectra
24
Frequency spectrum froma axial velocity signal in a turbulentaxisymmetric jet
It is always a good idea (DNS, LES, experiment) to check your specta.A -5/3 region should be evident if Re is high enough.
Resolution
25
Kolla et al, CST 2016
dth/Dx~40 (many points across flame)
hK/Dx~2 (finer than Kolmogorov)
Well-resolved DNS of premixed flames in Intense shear
Dissipation spectrum
Taylor hypothesis
26
If mean flow mainly in one direction, and 𝑘 (= 𝒖) ≪ ത𝑢, then:
Hence, a one-point temporal record is equivalent to a two-point measurement,and hence we can measure (“estimate”, rather) dissipation:
Also, integral timescale is 𝑇𝑡𝑢𝑟𝑏′ = 𝐿𝑡𝑢𝑟𝑏/ത𝑢
And eddy turnover time is 𝑇𝑡𝑢𝑟𝑏 = 𝐿𝑡𝑢𝑟𝑏/𝒖
Instrument response
27
How to estimate the fastest motion we may expect in an Eulerian measurement:
Hence, fastest frequency is U/hK , *NOT* 1/T’turb, 1/Tturb, or 1/tK.
uu
Lturb
hK
Mean flow U
u
time
Steep gradients
Mean flow U
Convergence of averages
Question: if I want to estimate the mean to a given accuracy, for how long should I be measuring?
UT: true mean ; Uest: estimated mean after N statistically-independentsamples; s2: true variance. Then:
28
T/T’turb : estimate of the number of statistically independent measurements over time period T in the turbulent signal with integral timescale T’turb. Error → 0 as N→ infinity. Error depends on s.
Millions of data points mean little, if not statistically independent.
“Flow-through times” – please avoid; it means almost nothing. Each point in your flow will converge differently.
Next few slides: some laminar flame concepts
- Autoignition with and without heat losses
- Well-stirred reactor
- Laminar strained diffusion flame
- Laminar strained premixed flame
- Conclusions we will draw:
- Introduce some concepts and quantities needed later
- balance of diffusion/convection and chemistry governs behaviour – how you
capture this phenomenon distinguishes turbulent combustion modelling
approaches.
29
30
Simple chemical model
1-step chemistry: F + n O ➔ (1+s) P
1kg F + S(=n MO/MF)kg O → (1+S) kg P
Assume constant cp, Le=1, D=constant, adiabatic. Then:
( )TREYYMWMW
MWAw oxfu
oxfu
fufu
02
/exp --=
fui
fu
ii
fui
fuw
x
YD
xx
YU
t
Y+
=
+
fuiii
ipp wQx
T
xx
TUc
t
Tc -
=
+
+
=
++
+
i
pfu
ii
pfui
pfu
x
TcQYD
xx
TcQYU
t
TcQY )()()(
QTTcYYTcQY
TcQYZ
pfufupfu
pfu
/)(
scalar conserved
00,00, --=+=
=+=
30
31
T
Reaction rate
TfTo
00, / TcQYT pfuf +=
)( TfCw fu =-
max,wT
0)( 0 Tw fu
Key conclusion: reaction rate very non-linear with T; has a maximum due to
coupling between T reached & conversion of reactants.
31
Simple chemical model
32
Autoignition without heat losses
Assume constant P, cp, Le=1, adiabatic, homogeneous.
Assume small reactants consumption until autoignition. Linearize reaction rate.
( )TREYYAMWMW
MW
c
Q
t
Toxfu
oxfu
fu
p
00,0,0
/exp -
=
B
( )
+D--=
D+-=
- .../1exp
)/1(expexp 0
00
0000
TTTR
E
TTTR
E
TR
E
-
-
-
20
00
000
)(exp expexp
TR
TTE
TR
E
TR
E
-
-=
2
00
0
00
)(exp exp
TR
TTE
TR
E B
t
T
)(exp -1 exp
12
00
0
00
20
0
--
=
TR
TTE
TR
E
E
TR
B t
exp
00
20
01
0,0,0
=
-
TR
E
E
TRYYA
MWMW
MW
c
Q oxfu
oxfu
fu
pign t"delay timeignition ")( =→Tt
32
33
Trends:
-As temperature increases, the ignition time decreases very fast.
-As P increases, tign decreases.
Realistic chemistry:
-Arrhenious behaviour often; complexities arise for heavy HCs
33
Autoignition without heat losses
34
Autoignition with heat losses
During induction time, heat losses to walls.
Ignition time longer than without heat losses
Time
Temperature
To
T, P
To
(a) (b)
LGTThcV
AwQ
ct
T
v
Vfu
v
-=--=
)(
10
34
35
Graphical solution:
T
Heat gain or loss
To
A
B
C
L1
L2
L3
G1G2
Excessive heat losses➔no ignition.
stable
Defines “Autoignition Temperature”.
It is NOT a fundamental property
unstable
35
Autoignition with heat losses
36
The Well-Stirred Reactor
“Longwell Bomb”
All quantities uniform inside, steady, constant P, adiabatic
Residence time=(volume)/(flow rate)
“out” = “inside”; different from “in”
Fuel+Air
Tin, YinProducts
Tout, Yout
T, Y
36
37
( ) VwYYm iiniouti ,, =-
( )TREYYMWMW
MWAw oxfu
oxfu
fufu
02
/exp --=
QTTcYY inpinfufu /)( , --=
QTTScYY inpinoxox /)(, --=
( ) VQwTTcm fuinoutp -=-
Species:
Energy:
( ) f(T) CTTV
min =-
m
Vtres
Heat Loss Heat Generation
37
The Well-Stirred Reactor
38
The Well-Stirred Reactor: extinction
( ) f(T) CTTV
min =-
unstable
stable
stable
TTfTin
Term in energy eq.
HT1
HT2
HTcrHT3
G
C
S1
S2
S3
S4
: realizable conditions
frozen (extinguished) burning
38
39
Extinction
Trends:
- Extinction due to a balance between residence time & kinetics.
- As Tin increases, stability improves.
- As tres decreases, T decreases and then suddenly extinction occurs
- Basic physics behind “S-shaped curve” of laminar flames
- Applications: WSR models used in GT modelling; flame blow-off
Fuel+Air
Flame
Recirculation zone
Treated as WSR
39
40
Laminar non-premixed (or diffusion) flames
• Laminar non-premixed flame structure not necessarily preserved in turbulent flames
• Key issues: location, extinction, radiation, CO, NO, soot
T, Y in mixture fraction space (will
be discussed in next slide)
40
Mixture fraction: key concept to understand non-premixed flames
• Conserved scalars (in 2-D for simplicity):
41
Mixture fraction
• The mixture fraction:
42
No fuel in air, no air in fuel
No source term!
• Stoichiometric mixture fraction: when Yfu=Yox/S.
• Knowledge of gives everything:
Discussed later (turbulent combustion)
Stream 1: “fuel”
Stream 2: “air”
43
Mixture fraction
• Mixture fraction is an extremely important concept in non-premixed & spray reacting flows:
• Gives mass fraction of fluid originated from fuel stream
• Covers the vast majority of practical problems (one-fuel combustion)
• Determination of mixing de-coupled from reaction (mixture fraction mixes in flow like
any passive scalar)
• For systems with more than one fuel or air stream, one can use “multiple mixture
fractions” (Fox’s book, 2003)
• Note: when differential diffusion effects are considered, different ways to define
mixture fraction become possible
43
44
Laminar diffusion flames: flame sheet model
• Flame pegged on stoichiometric iso-surface; reactants come to flame from either side; do not
penetrate; w=0 everywhere except at flame sheet.
Sandia D
Sydney Sandia
44
• Counterflow flame: a canonical problem highlighting the balance between fluid mechanics
(mixing, strain, scalar dissipation) and chemistry. Mixing controlled by strain rate A (equivalent
to scalar dissipation N=D[/x]2).
Max N across layer;
denote by N0
NN0
Double premixed
Non-premixed
Laminar diffusion flames: finite rate chemistry
45Model for scalar dissipation in flamelet and
CMC turbulent combustion methods
46
Laminar diffusion flames: finite rate chemistry
• Solve (i) either laminar diffusion flame in physical space and store results; or (ii) solve directly in
mixture fraction space
• Steady, transient
• Chemistry to any complexity can be used
• See Peters (2000)
• Structure of equations (const. cp):
46
Flamelet equations
Structure of strained non-premixed laminar flames• Finite rate implies: (i) overlap between fuel and air; lower T; curved corners. (ii) possibility of
extinction; (iii) needed for pollutants.
• Effects more pronounced as strain rate (or scalar dissipation) increases.
More leakage
at high strain
Borghesi, PhD, Univ. of Cambridge; using reduced CH4
mechanism from Massias et al, CTM, 1999
47
Structure of strained non-premixed laminar flames• As scalar dissipation increases, flame burns OK, but at reduced T and slightly reduced radical
levels.
48
Structure of strained non-premixed laminar flames• As scalar dissipation increases, some pollutants may change.
49
Structure of strained non-premixed laminar flames• Extinction scalar dissipation: denotes fastest possible mixing rate that the flame can survive.
Up to this point, flame burns, but at reduced T. After a critical scalar dissipation, no
combustion possible.
~400K
Extinction
GRI3, skeletal (Smooke),
modified 1-step (Q=f(f))
GRI 2.11 & reduced
Extinction
50
51
Further demonstration of Damkohler number effect
• Simple isothermal problem, reaction-diffusion balance. See overlap of reactants as Da
decreases.
0)1(
1)0(
1)1(
0)0(
2
2
1
1
212
2
2
2
212
1
2
1
==
==
==
==
-
=
-
=
hf
hf
hf
hf
ffh
ff
ffh
ff
Dat
Dat
51
52
Laminar counterflow premixed flame
• Solve with laminar flame code; then store results for use in turbulent flame model
• Some models only use SL
• Other use species mass fractions vs. progress variable c
• Other use wk(c)
52
Freely propagating flameStrained flame
(“back to back”)
53
Flame stretch
Flame area increases due to stretch. Stretch due to aerodynamic strain and curvature.
Local consumption depends additionally on Lewis number.
53From Cant & Mastorakos (2007)
54
Turbulent flame speed
Turbulent flame: more area than in laminar flames.
54From Swaminathan (Lecture notes) & Cant & Mastorakos (2007)
55
Turbulent flame speed
Turbulent flame: more area than laminar. Estimating area leads to ST model
55
Source: Shy et al., Proc. Roy Soc A (2000) 456:1997-2019
“Bending effect”
𝑆𝑇𝑆𝐿
= 1 + 𝑓(𝐿𝑡𝑢𝑟𝑏𝛿
)𝑢′
𝑆𝐿
𝑛
56
Turbulent premixed flame diagram
56
From Cant & Mastorakos (2007)
57
How does laminar flame respond to strain rate fluctuations?
57
Paxton, Giusti, Egolfopoulos, Mastorakos, CNF 2019
58
How does laminar flames respond to local extinction?
58
Paxton, Giusti, Egolfopoulos, Mastorakos, CNF 2019
Strain
time
extinction
59
How does laminar premixed flame respond to strain rate fluctuations andlocal extinction?
59
Paxton, Giusti, Egolfopoulos, Mastorakos, CNF 2019
As OH decreases, nothing left to consume CH2O, so
CH2O accumulates. Theoretical validation of the
conjecture of Kariuki et al. (PROCI 35) with CH2O-
PLIF in turbulent premixed flame.
60
Heat loss (conv / diff) vs. Heat generation
In both autoignition and WSR, the balance between heat loss (by convection or diffusion) and
heat generation (by chemical reaction) determines the nature of the solution.
In laminar non-premixed flames, reaction will cease if the strain rate A is too high. Reaction rate
can increase as A increases, but T decreases, and eventually extinction occurs.
Laminar flame simulations and experiments provide information on (i) chemical kinetics; (ii)
flame structure.
Both needed in turbulent non-premixed flame models, if these are targeting finite-rate
kinetic effects. If the turbulent flame simulation only targets flame length, perhaps
infinitely-fast chemistry is OK.
Similarly for premixed flames: in some turbulent flame models, we need only SL. In
others we need w(c), and we can choose from freely propagating flame, strained
single («burnt to unburnt»), or strained double («unburnt to unburnt») counterflow
flame.
60
61
Please do not use the phrase “turbulence – chemistry interactions” in vain…
For some, this term means only the large-scale flapping of a reaction zone that remains
unchanged from the laminar structure.
For some, this term means only the small-scale fluctuations of the mixing rate (e.g.
Kolmogorov-scale phenomena, strain fluctuations, dissipation rate fluctuations etc).
For some, this term means the statistics of the mixture generated before the flame.
My advice:
Specify which effect of turbulence you are talking about.
Think hard who your audience is – and what they are thinking and what their prejudices
are.
Better to discuss these “interactions” through an equation.
Distinguish between large-scale vs. small-scale, average vs. instantaneous.
61
Conclusions – Day 2
Basic turbulence ideas needed in turbulent combustion:
Reynolds decomposition
Diffusivity, range of scales, dissipation
Basic laminar combustion concepts needed for turbulent flames:
Balance of heat loss vs. heat generation
Structure of non-premixed flame
Structure of premixed flame
Some turbulent combustion models do not take local stretch effects into account; these
approaches may fail for finite-rate kinetic effects, local extinction, re-ignition, soot and other
quantities.
62